Section 4.3 Riemann Sums and The Definite Integral

Area under a curve The other day we found the area under a curve by dividing that area up into equal intervals (rectangles of equal width). This section goes one step further and says we can divide the area into uneven intervals, and the same concept will still apply. It will also examine functions that are continuous, but are no longer entirely non-negative.

The area under the curve on the interval [a,b] Area As An Integral The area under the curve on the interval [a,b] f(x) A a c

Distance As An Integral Given that v(t) = the velocity function with respect to time: Then Distance traveled can be determined by a definite integral Think of a summation for many small time slices of distance

If f(x) is a nonnegative, continuous function on the closed interval [a, b], then the area of the region under the graph of f(x) is given by This is our limiting process where we let n approach infinity 5

Notice that through the use of rectangles of equal width, we were able to estimate the area under a curve. This curve was always non-negative (i.e. it was always above the x-axis); therefore, the values we got when we multiplied f(xi) by Δx were always positive.

The graph of a typical continuous function y = ƒ(x) over [a, b]: Partition [a, b] into n subintervals a < x1 < x2 <…xn < b. Select any number in each subinterval ck. Form the product f(ck)xk. Then take the sum of these products.

The sum of these rectangular areas is called the Riemann Sum of the partition of x. LRAM, MRAM, and RRAM are all examples of Riemann Sums. The width of the largest subinterval of a partition  is the norm of the partition, written ||x||. As the number of partitions, n, gets larger and larger, the norm gets smaller and smaller. As n, ||x|| 0

Formal Definition from p. 303 of your textbook

Finer partitions of [a, b] create more rectangles with shorter bases.

Why is the area of the yellow rectangle at the end = b

We partition the interval into n sub-intervals Review a b We partition the interval into n sub-intervals Evaluate f(x) at right endpoints of kth sub-interval for k = 1, 2, 3, … n f(x)

This limit of the Riemann sum is also known as the definite integral of f(x) on [a, b] This is read “the integral from a to b of f of x dx,” or “the integral from a to b of f of x with respect to x.”

variable of integration (differential) Notation for the definite integral upper limit of integration Integration Symbol (integral) integrand variable of integration (differential) lower limit of integration

Definite Integrals vs. Indefinite Integrals A definite integral is number. An indefinite integral is a family of functions. They may look a lot alike, however, definite integrals have limits of integration while the indefinite integrals have not limits of integration. Indefinite Integra

Formal Definition from p. 304 of your textbook

NOW … Before we go any further, let’s make one quick – yet important – clarification regarding INDEFINITE INTEGRALS (with which we have worked before in finding antiderivatives) and DEFINITE INTEGRALS (which we are about to study now):

Today we will focus solely on the DEFINITE integral. The connection between the two types of integrals will probably not be clear now, but be patient. We will explore and examine the connection between the two next time. 

Evaluate the following definite integrals using geometric area formulas.

Top half only!

THEOREM: If f(x) is continuous and non-negative on [a, b], then the definite integral represents the area of the region under the curve and above the x-axis between the vertical lines x = a and x = b. a b

The Integral of a Constant

When functions are non-negative, the Riemann sums represent the areas under the curves, above the x-axis, over some interval [a, b]. When functions are negative, however, the Riemann sums represent the negative (or opposite) values of those areas. In other words, Riemann sums DO have direction and CAN take on negative values.

You try… Consider the function 2x2 – 7x + 5, find area, the rest is up to you I used 20 sub-intervals x = 0.1

Ex: Find the enclosed area with the x-axis and the function f(x) = x2 – 5x + 6 between the function’s roots, using 4 subintervals. Roots: x = 2, 3

Ex: First find the exact area enclosed by above the x-axis and the function. Then use Riemann sums with 4 and then 8 sub-intervals and left endpoints.

To summarize that thought … f A a b A1 f A3 = area above – area below a b A2